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Exponents are a mathematical shorthand to describe serial, or repetitive multiplication of the same number, like 2 x 2 x 2, or 5 x 5 x 5 x 5 x 5.
A number which is to be subjected to this repetitive multiplication, is called the base; the number of times it is to be multiplied by itself, is called the exponent. Multiplying a number in this way, is called "raising it to the nth power, where n = the exponent.
Baseexponent
Examples:
22 = 2 x 2 = 4
23 = 2 x 2 x 2 = 8
24 = 2 x 2 x 2 x 2 = 16
Special Cases - Exponents
- Exponent = 1. The value of any number raised to the 1 power, is always that number, e.g., 51 = 5
- Exponent = 0. The value of any number raised to the 0 power, is always 1, e.g., 30 = 1
- Exponent < 0. Negative exponents indicate a reciprocal value, e.g., 4-2 = 1/16
Special Cases - Bases < 0 and parentheses
Remember PEMDAS? Mathematicians have established the order in which we solve the different math operations in an equation. An easy way to remember the order is the acronym, PEMDAS.
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Order of Operations = PEMDAS
Here's the trick you need to memorize: P = Parentheses |
Problems involving order of operations are answered by first solving anything within parentheses, then dealing with exponents, then multiplication and division (working from left to right), and finally addition and subtraction. And if there are parentheses within parentheses, we work from the inside out.
PEMDAS is the key to correctly evaluating exponents where the base is a negative number.
Examples:
(-3)2 = -3 x -3 = 9
This makes sense, right? The quantity (-3) times itself = -3 x -3 = 9
-(3)2 = -(3 x 3) = -9
This makes sense, too. Using PEMDAS, we evaluate the exponential first: 32 = 3 x 3 = 9, then we apply the negative sign, which is really a shorthand way of saying we are multiplying by -1. So -1 x 9 = -9
-32 = -(3 x 3) = -9
What? Why is the answer negative nine, rather than positive nine? Again, we go back to PEMDAS, and evaluate the exponential first: 32 = 3 x 3 = 9, then we apply the negative sign, to get -9 for the answer.
Want more help with PEMDAS? Check out our order of operations worksheets and tips.
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